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Unformatted text preview: a n LTID system with transfer function H[z] t o a n everlasting
sinusoid o f frequency n is also a n everlasting sinusoid of t he s ame frequency. T he
o utput amplitUde is IH [ein]1 t imes t he i nput a mplitude, a nd t he o utput sinusoid is
shifted in phase with respect to t he i nput sinusoid by L H[e in ] r adians. T he p lot of
IH[einli vs n indicates t he a mplitude gain of sinusoids of various frequencies a nd is
called t he a mplitude response of t he system. T he p lot o f L H [e in ] vs n indicates t he
p hase shift of sinusoids of various frequencies a nd is called t he p hase response. T he
frequency response of a n LTID system is a periodic function o f n w ith period 21r.
T his periodicity is t he result of t he fact t hat d iscrete-time sinusoids w ith frequencies
differing by an integral multiple of 21r a re identical.
Frequency response of a system is determined by locations in t he complex plane
o f poles a nd zeros of its transfer function. We can design frequency selective filters by G
12 Frequency Response a nd D igital F ilters 778 p roper p lacement of its transfer function poles a nd zeros. Placing a pole (or a zero)
n ear t he p oint e jflo i n t he c omplex plane enhances (or suppresses) t he f requency
r esponse a t t he frequency n = no. Using this concept, a p roper c ombination o f
p oles a nd zeros a t s uitable locations c an yield desired filter characteristics.
Digital filters are classified i nto recursive a nd n onrecursive filters. T he d uration
o f t he impulse response of a recursive filter is infinite; t hat o f t he nonrecursive filter
is finite. For t his reason, recursive filters are also called infinite impulse response
( IIR) filters, a nd nonrecursive filters are called finite impulse response ( FIR) filters.
Digital filters c an process analog signals using A ID a nd D I A converters. P rocedures for designing a digital filter t hat behaves like a given analog filter a re discussed. A digital filter can simulate t he b ehavior o f a given analog filter e ither
i n time-domain o r i n frequency-domain. This s ituation l eads t o two different design procedures, one using a time-domain equivalence criterion a nd t he o ther a
f requency-domain equivalence criterion.
For recursive or I IR filters, t he t ime-domain equivalence criterion yields t he
i mpulse invariance method, a nd t he frequency-domain equivalence criterion yields
t he bilinear t ransformation m ethod. T he impulse invariance m ethod is h andicapped
b y t he aliasing problem, a nd c annot b e used for highpass a nd b andstop filters.
T he bilinear t ransformation m ethod, which is generally superior t o t he i mpulse
invariance m ethod, suffers from t he frequency scale warping effect. However, t his
effect can be neutralized by prewarping.
For nonrecursive or F IR filters, t he t ime-domain equivalence criterion leads t o
t he m ethod of windowing (Fourier series method), a nd t he f requency-domain equivalence criterion leads t o t he m ethod o f frequency sampling. Because nonrecursive
filters are a speci...
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